Invisible wrote:
>> For instance, if you consider polynomials over the real numbers modulo 
>> X^2-1, you get the same structure as the complex numbers (you can do a 
>> similar thing for quaternions).
> 
> Aren't quaternions non-associative?

Non-commutative you mean, but yes.  The quaternion representation using 
modular polynomials considers polynomials of two variables (x and y) and 
real coefficients where multiplication between x and y is 
non-commutative.  So the polynomials inherit their non-commutativity 
from their variables.

Note that it's not actually important to say what sorts of numbers x and 
y would actually represent -- we're just dealing with polynomials in 
their purest form, where *all* we know about x and y are they have 
operators like addition, subtraction, and multiplication which satisfy a 
few basic mathematical properties: 
http://en.wikipedia.org/wiki/Ring_(mathematics)

> ...and *this* is why I can't implement AES or CRC.

It's not so difficult as you might think, it's just takes some getting 
used to the math.  If you're interested in the implementation rather 
than the math end of things, Wikipedia has a typically verbose article 
on the subject: http://en.wikipedia.org/wiki/Computation_of_CRC

There's also some C code dealing with the polynomials used in AES at the 
bottom of this page: http://en.wikipedia.org/wiki/Finite_field_arithmetic


Also forgot to mention that (if I recall correctly) ideas related to 
this play an important role in some of the operations in computer 
algebra systems.

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